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The mechanics of the first bite

Kalpana R. Agrawal* and Peter W. Lucas

An analysis of the action of the incisor teeth in humans is presented in terms of the fracture of food
particles. It is predicted that the resistance of foods with an essentially linear elastic response to an initial
bite by the incisors will depend on the square root of the product of two food properties, Young’s modulus
and toughness. This quantity should be approximately equal to the product of the stress at cracking during
a bite, and the square root of the length of a notch or indentation from which that crack initiates. As a
test of the theory, the relationship between in vivo stresses and the depth of incisal penetration, measured
during bites on seven ‘snack’ foods by 10 subjects, and food properties established from mechanical
testing, was investigated. Theory and experiment were found to be in excellent agreement. A dimensionless
index of the efficiency of incision is suggested, relating fracture performance by subjects to values from
a testing machine. This appears to have a high level of inter-subject discrimination with efficiencies varying
about threefold. The method appears to have potential applications in dentistry, food science and studies
of human and primate evolution.Keywords: incisors; foods; fracture; bite mechanics

1. INTRODUCTION

Among living mammals, spatulate incisor teeth in both
upper and lower jaws are unique to the Anthropoidea, a
taxonomic group containing the apes, Old World and
New World monkeys. These teeth are relatively large in
fruit-eating primate species (Hylander 1975), making it
probable that the trait evolved in association with this type
of diet. However, humans use their incisor teeth on a
much more diverse range of foods than fruits in order to
control the size and shape of ingested food particles. The
teeth are of specific interest to researchers in two major
disciplines. A clinical speciality in dentistry called orthodontics
deals with the adjustment of tooth orientation.
Although the orthodontic treatment of incisors is usually
indicated by aesthetic considerations, functional analysis
suggests that a vertical incisal inclination, generally the
desired clinical outcome, may be out of alignment with
the general direction of muscular force (Paphangkorakit &
Osborn 1997). There have also been a large number of
studies by food scientists of incision related to consumer
preference. The acceptability of many classes of food to
consumers is strongly influenced by expectations about
food texture. Texture can be evaluated manually, but perceptions
concerning fracture involve the teeth and the first
bite with the incisors seems critical in formulating opinions
(Bourne 2002). Exactly how such sensations are
perceived is unknown and analyses have centred on the
correlation of psychophysical responses with the mechanical
characteristics of foods, the latter as obtained from
often rather arbitrary tests on universal testing machines.
Significant correlations between a perceived food quality
and a mechanical property should not imply a functional
relationship because if only a limited range of foods is
offered, their mechanical properties would be likely to be
correlated with each other (Ashby 1998). Spurious correlations may follow, which may well break down over a
broader range of foods.

Most uses of the teeth involve the fracture of foods.
However, food scientists have mainly ignored the theory
of fracture in their mechanical characterizations. Applying
fracture mechanics, Vincent et al. (2002) found that the
critical stress intensity factor, KIC, of foods, a parameter
related to crack initiation, was linearly related to perceived
measures of ‘hardness’ and ‘crunchiness’ as evaluated by
trained ‘taste’ panellists. This paper explores this fracture
criterion further by comparing it with in vivo conditions,
the aim being to provide a theoretical basis for incision
and an index of dental efficiency that could be used for
studies of incisor function in humans.

2. THEORETICAL BACKGROUND

Usually, in an incisal bite, a hand holds a fruit like an
apple close to the edges of the upper incisors while the
lowers are swung upwards towards it causing, first, an
indentation on the lower surface (figure 1a), followed by
cracking (figure 1b). The upper incisors may also move
due to the action of the neck muscles, which can influence
the food surface on which a crack is initiated (Osborn et
al. 1987). Our analysis is close to that of Osborn et al.
except that it is not based on fracture stress, but on energy.
Any fracture event can be analysed in two ways: (i) in
terms of the balance between the strain energy within the
particle and the energy required to extend the crack from
the indentation; or (ii) in terms of the effect on the stresses
in the apple caused by the presence of the indenting incisors.
We can model this generally only for a quasi-static
elastic fracture, in which the fractured surfaces still fit
together to resemble the original and where the crack
growth is relatively slow. The propagation of a crack
initiated from the indentation formed by the lower incisors
will be at a critical value of the Irwin stress intensity, KIC,
given by where the fracture stress is sF , C1 is a constant and a is
the depth of indentation of the lower incisor (Atkins &
Mai 1985). Provided the stress–strain curves of foods are
essentially linear, then another way of expressing the same
result is the use of the Griffith equation for the energy
balance within the apple tissue

Figure 1. (a) A food particle (grey) is held against the upper incisors. Movement of the lower incisors relative to the food
produces an indentation from which, in (b), a crack eventually initiates. In (c), the experimental set-up is shown. Stresses are
registered on the upper face of the food particle, remote from the crack, by a two-layered pressure-sensitive film. The film is
covered in plastic wrap and both food particle and wrapped film are then enclosed in a 0.4 mm thick latex finger-stall. Posttest
measurement of the penetration of the lower incisor was aided by painting either the lower surface of the food particle or
the inside of the finger-stall with lipstick.

where E is the Young’s modulus and R is the toughness
of the apple tissue (Atkins & Mai 1985). C2 is another
dimensionless constant where, in idealized loading conditions
in which the load is applied remote to a crack in
a large body, C1 = C2 = π0 .5 . These conditions hardly apply
to the action of the teeth and, therefore, in combining
equations (2.1) and (2.2), we must define a new constant
depending on the loading geometry:

If, on the one hand, we can measure the stresses in the
food particle at cracking in vivo and measure the food
properties, Young’s modulus and toughness, with mechanical
testers, then we can use a fourth relationship

to test whether KIC forms an effective criterion for crack
growth during an incisal bite. The material properties on
the right-hand side of equation (2.4) can be measured easily
on a universal testing machine. The issue is whether
the factors on the left-hand side, the in vivo stress at cracking
and the depth of indentation, can also be measured.
If so, then a straight-line relationship is predicted, passing
through the origin, with the dimensionless constant, C4 ,
being given by the slope of a graph where σFa0 .5 is plotted
against (ER)0 .5 . This constant offers a measure of incisal
efficiency, defined as the ratio of in vivo fracture to ‘in
vitro’ values obtained from machine tests. The lower the
value of C4 , the greater is the efficiency of the lower incisors.
The efficiency of the upper incisors could be tested
separately if conditions can be organized such that they alone penetrate the food particle to cause cracking
directed from its upper surface.

3. MATERIAL AND METHODS

Foods were four types of cheese, raw carrot, Brazil nut and
macadamia nut kernels (in table 1). The toughness, R, of the
cheeses and carrot were obtained with a 15° included angle
sharp stainless steel wedge driven into a rectangular block of
food (Khan & Vincent 1993) at a crosshead speed of
140 mm min21. The test is shown schematically in figure 2a.
From point A, the force built with indentation until cracking at
B. The crack was unstable at first but quickly settled (point C).
Further movement of the wedge between C and D produced a
stable crack running just ahead of the wedge tip. The area under
the curve was measured between C and D. The wedge was then
reversed to position A and run through the cracked specimen
producing a friction curve labelled AEF. The mean friction
varied between 5.8% (carrot) and 25.0% (reduced fat cheddar)
of the total work done. The area bounded by CDFE gave the
work done to produce the crack. Nut kernels were too brittle
for crack growth to be controlled in this test. Therefore, the nuts
were sectioned to a thickness of 0.15–0.88 mm and then cut
using scissors (Dovo, Germany) using a portable tester (Darvell
et al. 1996) equipped with a motor producing cutting speeds
of 30–50 mm min2 1. The tests, shown in figure 2b, showed no
apparent dependence of toughness on section thickness, but
both nuts were tested over a similar thickness range as a precaution.
Only tests in which the crack was controlled to the end
of the specimen were classed as valid. The Young’s modulus, E,
of the foods was obtained by compressing cylindrical or cuboid
specimens (made with cork borers or dies) of low aspect ratio
on a Lloyd LRX (UK) universal testing machine at a crosshead
speed of 5 mm min2 1. A thin adhesive-backed Teflon strip
attached to the compression plates served to reduce friction.

For the bite tests, foods were shaped into rectangular blocks,
measuring 12 mm (height) ´ 12 mm ´ 20 mm for cheeses and
carrot or 8 mm (height) ´ 10 mm ´ 14 mm for the nuts. The
12 mm width of the block (figure 3a) was considerably smaller than the sum of the mesiodistal length of the four lower incisors,
averaging ca. 20 mm.

Table 1. Statistics of the in vivo data regressed against food properties from machine tests for individual subjects.

Figure 2. (a) Diagram of the configuration of a wedge test, designed to estimate the toughness of specimens of cheeses and
carrot. The test involved two passes of the wedge, one to initiate and grow a crack just ahead of the wedge tip (marked by
ABCD) and the second, to deduct the work of friction by passing the wedge against the fractured surface (AEF). The length
of the crack could be estimated from the displacement of the wedge. The area CDFE on the force–displacement graph
encloses the work related to the fracture and corresponds to the fractured area (bt) shown in black at left. (b) Diagram to
show an analogous test with scissors on thin sections of nut kernels. The graph area ABCED gives the work done. Specimens
were cut through completely to simplify measurement of the crack area. For both tests, the work done divided by the crack
area gave toughness.

Ten subjects (five males, five females, aged 22–49 years, mean
of 31 years, s.d. of 8 years) made one bite on a standard sized
food particle enclosed in a 0.4 mm thick latex finger-stall (a latex
finger-tip protector). The finger-stall shielded the food’s identity
from subjects and prevented salivary contamination (Agrawal et
al. 1997; Lucas et al. 2002). Subjects were asked to pick the
particle up by hand, place this between the incisors and bite
normally, but to stop just as a particle ‘gave way’, i.e. cracked
(figure 1c). This last part of the instruction was required to avoid
the recording of stresses at tooth–tooth contact, which could
easily have exceeded the stresses during food loading. No subject
had difficulty with this instruction and the teeth did not carry
through to tooth–tooth contact. The cracks in three of the
cheeses and raw carrot usually arrested within the food block,
but those in Parmesan cheese and nut kernels continued in order
to subdivide the particles.

Kohyama & Nishi (1997) pioneered in vivo measurement of
the pressure distribution on food surfaces during incision. The
sheet sensors they used are very thin and flexible (less than
0.1 mm; Kohyama et al. 1997), but produce a continuous electrical
signal that can be converted to pressure via calibration
curves, provided the sheet remains flat during a bite. Instead,
being interested only in the conditions at fracture, we employed
a pressure-sensitive film (Fuji Prescale, Japan) that is thicker and
stiffer than a sheet sensor, but can only record the maximum
compressive stress during the test. These films were cut to the size of the food block and placed on its upper surface. The films
come in two 90 m m thick sheets (i.e. 0.18 mm in total), called
‘A’ and ‘C’, which release a red dye when pressed together. In
the tests, the films were wrapped in a single layer of 10 m m thick
plastic food wrapping to avoid damage by fluids (Liggins et al.
1995). For some foods, a coating of dark lipstick was painted
on the inside of the finger-stall lying against the lower food surface
or on the food surface itself (figure 1c). This served to mark
the depth of incisal penetration prior to cracking. Each subject
repeated the test once on each of the seven foods, which were
given in random order. Afterwards, both the average maximum
compressive stress against the incisors and the depth of incisal
indentation were measured using an image analyser.

The films require user calibration because the extent of colour
development, depending basically on the compressive stress,
also varies to some degree with temperature, humidity and the
rate and duration of loading (Liggins 1997). Temperature and
relative humidity in the laboratory were controlled at 21 °C and
55–70%, respectively. The exact loading conditions during a
bite probably varied between subjects and we did not measure
the velocity of jaw movement in this experiment. Judging from
the experiments of Kohyama et al. (2001), incisal velocities may
reach 10 mm s2 1. However, we standardized conditions to a
rapid loading rate held momentarily. We pressed a hemispherical
indenter, attached to the crosshead of the universal testing
machine, onto the films, moving it at 900 mm min2 1 to its target.
We then stopped the crosshead to make a static indentation
held for ca. 1 s duration at a variety of loads (measured by 10 or 50 N load cells). The indenter was then unloaded at the same
crosshead speed and indentations measured with a Leica QWin
(UK) image analyser. The surface area of the indentation was
measured and divided into the load to obtain the average pressure.
The optical density of dye release was obtained by reading
the grey level obtained from image analysis. A log–log plot of
grey level versus average pressure was linear within bounds close
to those indicated by the manufacturer. The stress range for a
film was, however, not much greater than an order of magnitude
and two films with overlapping stress ranges, ‘low pressure’ and
‘medium pressure’, were required.

Figure 3. (a) Penetration of ‘extra sharp’ Cheddar produced by two central lower incisors, shown in top view. The part of the
specimen to the right side was inserted in the mouth: the ‘shovel-shaped’ nature of the incisal edge, common in ethnic
Chinese, is clearly seen. The lipstick stain shows the depth of indentation. Scale bar, 10 mm. (b) Colour development (shown
as a grey area) of a ‘low pressure’ film produced during loading by one of these incisors. Scale bar, 1 mm. (c) A side view of
the Cheddar specimen showing the depth of penetration, confirmed by inspection of lipstick staining, marked as a. A crack is
seen extending from this indentation. Scale bar, 10 mm.

After each experiment, the finger-stall was opened and the
films subjected to image analysis as described above. The average
optical density was measured, from which the average compressive
stress could be obtained by reference to the standard
curve.

4. RESULTS

No surface slippage between food particle, film and
finger-stall was observed during the tests. The extent of
incisal penetration (figure 3a,b) was usually clear. The
shape and extent of the red stain on the pressure-sensitive
film was consistent with incisal morphology (figure 3a).
Figure 4 shows the mean for all subjects of in vivo stressrelated
measurements on the y-axis regressed against
instrumental (in vitro) evaluations on the x-axis. A linear
relationship, obtained by least-square regression, is clearly
seen (r2 = 0.986; p < 0.001). The mean value of the slope,
C4, was 14.64 (s.e. of 0.793). The y-intercept,
81.02 kPa m1 /2 (s.e. of 36.73), was not significantly different
from zero ( p<0.08). Each point in figure 4 was subject
to considerable inter-subject variation (shown by the
s.d.), but each subject produced a linear response (table
1). For the mean of the two attempts by each subject,
there was a significant relationship (r2 ranged between
0.70 and 0.97; p<0.02 or better) and the value of C4
varied approximately by a factor of three (table 1). The
mean toughness and Young’s moduli of the foods, shown
in table 2, were not correlated with each other
(r = 20.02; p >0.9).

5. DISCUSSION

It appears that the theory of fracture applies just as well
to a bite by the incisors at the entrance to the mouth as it does to the physical world away from the lips. The critical
stress intensity factor, KIC, or its energetic equivalent
(the square root of Young’s modulus times toughness),
appears to be an appropriate characterization for these in
relation to incision, fully supporting the psychophysical
analysis of Vincent et al. (2002). Important features of
foods are that their stress–strain curves are close to linearity.
Nut kernels have a linear response and fail at small
strains. In some of the other foods, the slope can decrease
at higher strains, but it is very difficult to judge whether
this is due to fracture (Charalambides et al. 1995). Carrot
and nut particles fracture rather than yield. However, the
fractured surfaces of foods, beyond the point of incisal
indentation, did not show significant plastic distortion
(figure 3c), implying that the assumptions made here are
appropriate. Nevertheless, investigations on a broader
scale could benefit from the nonlinear analysis developed
by Purslow (1991) because this allows results from linear
and nonlinear behaviours to be compared.

Vincent et al. (2002) suggest that mechanical characterizations
of foods founded on sound theoretical support
could replace the need for taste panels and psychophysical
correlations in food texture studies, since mechanical tests
are simpler and cheaper to run than taste panels and are
free from cultural and linguistic limitations. However, it
needs to be established that these tests do indeed match
fracture conditions in the mouth. We argue that physiological
investigations, such as those reported here, provide
necessary in vivo confirmation. These experiments are so
basic that food scientists could use the efficiency index to
map the ease of biting on new products, even as part of
a quality control regime. The experiments could be performed
without a finger-stall, but it is difficult to preserve
hygiene in experiments like this, even though the pressuresensitive
films can apparently be sterilized (Liggins et al.
1994). The opacity of the finger-stall shielded subjects
from recognizing the food visually and from any disturbing
influence following flavour release after the bite, but a
thinner finger-stall is perhaps indicated.

Dentists might use the method to establish the anatomical
and physiological factors that influence the incisal
efficiency index. The experiments here are relevant mainly
to the lower incisors, but the procedure could be reversed
with the film placed on the lower surface of the food particle.
Then, with the particle held just above the lower
incisors, the head could be moved to make a bite with the upper incisors. Thus, two efficiencies could be measured,
one for the upper and the other for the lower incisors.
However, establishing the parameters that influence these
efficiencies requires a large sample of subjects drawn from
a population, making impressions of the incisors and
establishing their orientation. The combination of morphological
factors that produce the lowest value of C4 , i.e.
the lowest multiple of that which testing machine values
generate, may define the optimum functional state. Such
an investigation must surely be of significance in guiding
the philosophy behind orthodontic treatments. The
measurement of physiological variables must also play a
part in such a survey. Prime among these must be the
velocity of jaw movement and the strain rates that could
be imposed on food particles. These are very difficult to
investigate during mastication with the post-canine teeth,
although the relatively invariant rhythm of jaw motion
suggests that smaller particles suffer higher strain rates
than larger ones. The strain rate may, of course, not have
any general relationship to the speed of cracks that form
as particles become fragmented (Lucas et al. 2002). However,
all these factors are far more amenable to analysis
during a first bite and could help to establish whether the
quasi-static conditions assumed here are violated.

Table 2. Means and standard deviations of mechanical measurements on the test foods.
(The sample size was 10 for each test.)

Figure 4. The mean for all subjects of the product of the
average maximum in vivo compressive stress during incision
and square root of penetration distance of the lower incisors
is plotted on the y-axis against the square root of the
product of Young’s modulus and toughness of food (both
measured on a mechanical tester). A linear relationship,
predicted in the text, is shown. Mean for all subjects
r2 = 0.986; p<0.001.

The incisors of anthropoid primates vary greatly in their
size, shape and occlusal relationships. There are several
means by which mammals in general (Osborn et al. 1987), and primates in particular (Ungar 1992), ingest foods. Several
of these involve the use of the teeth more as a way of
gripping foods than as a direct means of fracture (figure 5).
An example of this in primate feeding is ‘stripping’
(Osborn et al. 1987; Ungar 1994), which involves pulling
leaves off a branch still attached to its parent plant. Primates
also use their hands extensively when feeding,
allowing relative movement between the hands and incisor
teeth to produce some element of tension. The experiments
reported here probably apply most closely to behaviour
with fleshy fruits, particularly ‘peeling’ (Gautier-Hion
et al. 1985; Leighton 1993) and ‘scraping’ categories.
However, if all such food processing methods could be
mimicked, then in combination with field measurement of
the mechanical properties of these foods by primatologists
(Lucas et al. 2001), studies on humans might have a critical
part to play in explaining the evolution of incisor shape.

Figure 5. The various modes of ingestion employed by
primates. Some involve the use of incisors to crack objects,
while others involve them in grip. (Adapted from Osborn et
al. (1987) and Ungar (1992) with permission.)

The authors thank the Committee for Research and Conference
Grants (University of Hong Kong) for financial support.